Abstract

In recent years, the Krasnoselskii fixed point theorem for cone maps and its many
generalizations have been successfully applied to establish the existence of multiple
solutions in the study of boundary value problems of various types. In the first part
of this paper, we revisit the Krasnoselskii theorem, in a more topological perspective,
and show that it can be deduced in an elementary way from the classical
Brouwer-Schauder theorem. This viewpoint also leads to a topology-theoretic generalization of
the theorem. In the second part of the paper, we extend the cone theorem in a different
direction using the notion of retraction and show that a stronger form of the often cited
Leggett-Williams theorem is a special case of this extension.